Axiom Of Choice

This non-implication, Form 305 \( \not \Rightarrow \) Form 253, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6875, whose string of implications is: 164 \(\Rightarrow\) 91 \(\Rightarrow\) 305

A proven non-implication whose code is 5. In this case, it's Code 3: 444, Form 164 \( \not \Rightarrow \) Form 253 whose summary information is:

Hypothesis	Statement
Form 164	<p> Every non-well-orderable set has an infinite subset with a Dedekind finite power set. </p>

Conclusion	Statement
Form 253	<p> <strong>\L o\'s' Theorem:</strong> If \(M=\langle A,R_j\rangle_{j\in J}\) is a relational system, \(X\) any set and \({\cal F}\) an ultrafilter in \({\cal P}(X)\), then \(M\) and \(M^{X}/{\cal F}\) are elementarily equivalent. </p>

This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 305 \( \not \Rightarrow \) Form 253 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name	Statement
\(\cal N3\) Mostowski's Linearly Ordered Model	\(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\)

Implications